Download Experiment 13 Elastic Potential Energy of a Stretched

PHYSICS EXPERIMENTS — 131
13-1
Experiment 13
Elastic Potential Energy of a Stretched Object
The purpose of this experiment is to determine
the work done in stretching an elastic object. This
work done may, alternatively, be thought of as an
increase in elastic potential energy stored in the
object. This determination is then used to make
predictions on the vertical motion of the stretched
object, utilizing the concept of transformation of
energy from elastic potential to gravitational
potential.
Preliminaries.
Stretching any object, whether a rubber band
or a steel spring, requires a force felas. Application
of this force over a distance results in work W done
to store energy as elastic potential energy, Uelas, in
the stretched object. If the object is stretched from
a length X0 to a length X, then
W = U elas =
at the bottom. As the hanging mass is not moving,
felas is simply equal to the hanging weight Mg by
Newton’s First Law. The stretch y = X−X 0 is
determined as M is varied and the data represented
in a graph of felas versus y. See Figure 2, in which a
smooth curve has been fitted to the individual data
points.
Typically, this graph will have a linear section
near the origin, (once the slack is taken out of the
rubber). This is known as the elastic region, where
Hooke's Law holds for small elongations.
f
elas
X
∫
f elasdx
X0
In any real-world system, there is no simple
formula available to describe the stretch force and
an analytic solution to the integral is not possible.
You will measure and graph the stretch force felas on
a thick rubber band as a function of the stretch
distance. The elastic potential energy can then be
determined from the area under the curve of this
graph.
The technique for measuring the stretch force
is shown in Figure 1. The rubber band is suspended
vertically and a mass M is hung
(a)
(b)
Ball
Xo
X
M
Figure 1. Determination of elastic force
y
Figure 2. Elastic force vs stretch of an elastic cord
The area under the curve between y=0 and
y=L, where L is the maximum amount that the
object is stretched can be determined in many ways.
One method is to approximate the area by a series
of rectangles. The range from y = 0 to y = L is
divided into rectangles of equal width Δy. Example:
if L = 55 cm, then 11 rectangles could be used since
this gives each rectangle a simple width of Δy =
55cm/11 = 5 cm. The height of each rectangle is
picked to give a reasonable fit to the real area. The
height has units of force and is read off the vertical
axis: f1, f2, f3, ... Some rectangles are shown in
Figure 3.
13-2
PHYSICS EXPERIMENTS —131
f elas
Procedure.
f4
f3
f2
f1
Δy
Δy
L
y
.
Figure 3. Approximating the area under the curve
The height of each rectangle is fi and the area of
each rectangle is fiΔy. The total area is Σ(fiΔy) =
ΔyΣfi. We have “numerically integrated” the curve
to yield Uelas = ΔyΣfi. Remember, the fi are read
from the graph, and are not the measured data
points of felas.
The rubber band may be used as a slingshot. If
it is held vertically, stretched an amount y=L and
suddenly released, then the rubber band is capable
of projecting a mass up into the air. In this
projection, some of the stored Uelas is converted
into kinetic energy of the projected object. This
kinetic energy gets converted into gravitational
potential energy as the object rises into the air and
slows. The gravitational potential energy is given
by
Ugrav = mgh,
where m is the object mass, g is the acceleration due
to gravity, and h is the maximum height reached by
the object above the launch point.
Exactly how high the projected object flies
depends on more than just the original elastic
potential energy. The maximum height attained also
depends on the retarding effects of air friction, the
exact manner in which the rubber band is released,
energy lost to heat in stretching the rubber band,
and a host of other effects. These effects are
difficult to account for, but they all act to reduce the
amount of energy transferred to gravitational
potential energy by an amount Wloss.
Conservation of energy gives
mgh = Uelas - Wloss.
You will determine the rise height h to determine
Wloss .
• Put the band over the corner of the ruler, pull
back, and release so that the ball at the end of the
band is launched vertically. Then by trial
determine how much you should stretch the band
so that the ball just makes it to the ceiling.
Record the distance L band had to be stretched to
reach the ceiling and measure the height h from
the initial position of the ball to the ceiling.
• Mount the rubber band vertically. Determine
and record its relaxed, unstretched length X0.
Now suspend some mass M from the band and
determine the stretched length X. Repeat for
increasing values of M until the band is stretched
the same amount L that you had when it was
projected and just made it to the ceiling. Record
the data in a table.
• Compute the stretch force felas and the stretch y.
Add these values to the table.
• Make a plot of felas vs y. Find the work W=Uelas
done in stretching the band a distance L by
determining the area under the curve.
• Predict how high H the object should have flown
if there were no energy losses, Wloss = 0.
• Ignoring Wloss is unrealistic. Use conservation of
energy to determine Wloss and the fraction f of the
original energy that does not go into gravitational
potential energy.
f = Wloss/Uelas = (Uelas-mgh)/Uelas
{Show that f = 1- (h/H).}